Block designs with $\gcd(r,\lambda)=1$ admitting flag-transitive automorphism groups

TL;DR

This paper classifies 2-designs with gcd(r,λ)=1 that admit flag-transitive automorphism groups, identifying known examples and cases where the design has q=p^d points with specific automorphism group structures.

Contribution

It provides a classification of such 2-designs, expanding understanding of their automorphism groups and identifying new structural conditions.

Findings

Known examples of 2-designs with flag-transitive automorphism groups are characterized.

Designs with q=p^d points have automorphism groups as subgroups of AΓL_1(q).

The paper narrows down possible automorphism group structures for these designs.

Abstract

In this paper, we present a classification of $2$-designs with $\gcd(r,\lambda)=1$ admitting flag-transitive automorphism groups. If $G$ is a flag-transitive automorphism group of a non-trivial $2$-design $\mathcal{D}$ with $\gcd(r,\lambda)=1$, then either $(\mathcal{D},G)$ is one of the known examples described in this paper, or $\mathcal{D}$ has $q = p^{d}$ points with $p$ prime and $G$ is a subgroup of $A\Gamma L_{1}(q)$.